What new beginnings do your characters experience? Do they seek these or have to make the best of a bad job, so to speak, because said new beginnings are imposed on them? Are there new beginnings they want to have but so far are eluding them?
How do the new beginnings change them (and there is bound to be some change)? (Best example of this is Frodo in The Lord of the Rings. Completely changed by his new beginning, his adventures). Do others around them resent the new beginning because they themselves are being left behind (figuratively, not necessarily literally)?
And why do you want your characters to have new beginnings? How do they advance the plot? What more do they reveal about the characters that we need to know?